Tips

It is important to note the difference between the differentiation with $d$ and $\partial$ symbols. Sometimes these symbols are used interchangeably, but this is a wrong practice.
Let's look at the following example:$f(x,y) = x^2+3xy$, where $y=x^3$. Then $\frac{\partial f}{\partial x} = 2x+3y = 2x+3x^3$, but $\frac{df}{dx} = \frac{d(x^2+3x^4)}{dx} = 2x+12x^3$. The same derivative can be computed by: $\frac{df}{dx} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}= 2x+12x^3$

The notation of higher order derivatives $f^{(n)}$, namely $\frac{d^n f}{dx^n}$, is not just a strangely chosen symbols to keep us clueless why d is in power $n$ in one place and $x$ in another (actually it is $(dx)^n$ since $dx$ is a single impartible symbol for differential). The explanation is simple, we can think on the derivative as a quotient of two differentials, and it can be proved that $d^n f = f^{(n)}(dx)^n$ (see differential).

The differential notation is particularly convenient for understanding integration. You can think on an integral as a sum of function values multiplied by differentials of independent variable (infinitely small segment, where all segments sum up to the entire region of integration). Formally, this approach is wrong mathematically, but should suffice for us, engineers.